New sharp lower bound for the quorum coloring number of trees - Sahbi, Rafik (Information processing letters, 2022)
•The alliance partition number of a tree has a new sharp lower bound computable in linear time.•This new lower bound is better than all those that have been previously established.•There is a relationship between the order, the diameter, the vertices degree and the matching number of a subgraph of a tree.•The alliance partition number of a binary tree is computable in linear time. A partition π={V1,V2,...,Vk} of the vertex set V of a graph G into k color classes Vi, with 1≤i≤k is called a quorum coloring if for every vertex v∈V, at least half of the vertices in the closed neighborhood N[v] of v have the same color as v. The maximum cardinality of a quorum coloring of G is called the quorum coloring number of G and is denoted by ψq(G). A quorum coloring of order ψq(G) is a ψq-coloring. In this paper, we partially answer an open problem concerning quorum colorings of graphs. Namely, we improve a sharp lower bound given in 2012 by Eroh and Gera on the quorum coloring number of a nontrivial tree, and show that our new lower bound can be computed in linear time. Moreover, we show that this bound is attained by all non trivial binary trees.